TfL is actually already part funded via business rates and council tax, via the GLA. This arrangement replaced the previous one, where it got a grant from central government instead, just a year or two before Covid hit.
As i understand it, TfL is quite unusual for a big-city public transport system in being quite heavily reliant on ticket revenue for its funding (about 50% I think). In other countries, a much lower proportion of funding tends to come from farebox revenue.
This is why it was so badly hit by the drop in usership during Covid, and has had to rely on emergency bailouts from central government to compensate.
Also, because it's responsible for London's trunk roads network, it means that their upkeep is funded via business rates / council tax / public transport fares, and not "road tax". So, drivers' claims that VED pays for the upkeep of roads that pedestrians, cyclists and buses dare to use have even less validity in London - effectively none of their VED now comes back to TfL because it goes to central goverment. If they pay congestion charge, some of that does come back to TfL but otherwise their private motoring around London is subsidised by business rates and council tax paid by a population many of who don't own a car - and by public transport users.
On the question of whether public transport should be free - in London it would mean a need to at least double the amount of "subsidy". I don't see why it should be all or nothing - you could have a discount for low income Londoners and still extract revenue from tourists and well-off commuters. And because London does have capacity problems, there's probably a good argument against making it completely free, where the risk is not that people transfer from car to public transport, but from walking/cycling, on journeys or journey legs where it's perfectly feasible to do that.
I think they found that this switch from walking to public transport was one of the effects of making it free in Tallinn. I'm not any expert on what they did there but you might find some significant differences from London - like the extent to which fares were already subsidised, or the sheer number of people that need to be moved, compared to the system's capacity. One of the features of London is that it needs to shift an enormous number of daily commuters from outside London, on top of the resident population.
Also, I don't think Tallinn just made public transport free without doing anything else - they changed parking fees, re-allocated road space and so on. In other words they did a lot of the stuff that some people like to think free transport could be instead of rather than as well as.
If you were to make public transport free in London, then it probably would be inevitable that you would increase bus capacity, because there's little space to increase it on tube/rail. And to do that you'd probably need to increase the allocation of road space for buses, whether the new bus users were switching from car or from walking, or just making journeys they wouldn't have made otherwise.